On the Completeness of Context-Sensitive Order-Sorted Specifications

We propose three different notions of completeness for term rewrite specifications supporting order-sorted signatures, deduction modulo axioms, and context-sensitive rewriting relative to a replacement map μ. Our three notions are: (1) an appropriate definition of μ-sufficient completeness with respect to a set of constructor symbols; (2) a definition of μ-canonical completeness under which μ-canonical forms coincide with canonical forms; and (3) a definition of semantic completeness that guarantees that the μ-operational semantics and standard initial algebra semantics are isomorphic. Based on these notions, we use equational tree automata techniques to obtain decision procedures for checking these three kinds of completeness for equational specifications satisfying appropriate requirements such as ground confluence, ground sort-decreasingness, weakly normalization, and left-linearity. Although the general equational tree automata problems are undecidable, our algorithms work modulo any combination of associativity, commutativity, and identity axioms. For all combinations of these axioms except associativity without commutativity, our algorithms are decision procedures. For the associativity without commutativity case, which is undecidable in general, our algorithms use learning techniques that are effective in all practical examples we have considered. We have implemented these algorithms as an extension of the Maude sufficient completeness checker.